As an example, and also because it even has some
practical importance, let's look at one kind of transmission
line. It is called a stripline and it looks like
Figure 1. It consists of a flat conductor, located
between two ground planes. It is supported by an insulating
dielectric with dielectric constant
εε. This is kind of like
the situation you would find on a multi-level PC board, where
perhaps the bus lines would be running on an inner layer with
ground planes above and below them.
Between the center conductor and the ground plane, there will be
some capacitance,
CC. If we can
assume that the electric field is more or less confined to the
regions between the strip conductor and the ground plane (which
occurs when the ratio of
WB
W
B
is not too small) then for either capacitor (assuming
unit length into the picture) we will get a value
C=εWB2
C
ε
W
B
2
(1)
since the value of a capacitor is just the dielectric constant
times the area of the plates, divided by the spacing of the
plates.
Looking quickly at Figure 1 you
might think the two capacitors are in series, but you would be
wrong! Note that each capacitor has one lead connected to the
center conductor and the other lead connected to ground, and so
the two capacitors are in fact, in parallel, and hence their
capacitances add. Thus, for the capacitance per unit length for
this line, we can write:
C=4εWB
C
4
ε
W
B
(2)
It can be shown (although we won't do it here) that for
any transmission line where the electric
and magnetic fields are perpendicular to one another (called
TEM or
transverse electromagnetic) the
speed of propagation of the wave down the line is just
v
p
=cε
ε
0
=3×108ms
ε
r
v
p
c
ε
ε
0
3
10
8
m
s
ε
r
(3)
Where
ε
r
ε
r
is called the
relative dielectric constant
for the material. Well, we also know that
v
p
=1LC
v
p
1
L
C
(4)
From which we can write
L=1
v
p
2C=B
v
p
24εW
L
1
v
p
2
C
B
v
p
2
4
ε
W
(5)
We can now insert this value for
LL
into the expression for
Z
0
Z
0
, the impedance of the line.
Z
0
=LC=B
v
p
24εW4εWB=B4εW
v
p
=B4εWc
ε
r
Z
0
L
C
B
v
p
2
4
ε
W
4
ε
W
B
B
4
ε
W
v
p
B
4
ε
W
c
ε
r
(6)
And so, we have derived an equation for the impedance
Z
0
Z
0
of the line in terms of the dimensions
WW and
BB, the dielectric constant of the
insulating material,
εε,
and
cc, the speed of light. How
good is this expression, and in particular how good is our
assumption that the electric field is all confined to the region
under the conductor? Not so great actually
Figure 2.
Figure 2 shows the results from using
Equation 6 and a more exact calculation, which takes into
account the fringing fields. As you can see we have to get the
ratio
WB
W
B
up to about 4 or so before the two match. But at least
we get the right behavior and we're not totally out of the ball park.
